# Canonical reference for Chern characteristic classes

I'm a little uncertain about the definitions for

1. Chern roots

2. Chern classes

3. Chern characters

From perusing several discussions, I gather that if one correlates the nomenclature with that of symmetric polynomials/functions and their relationships to characteristic polynomials of a generic square matrix $$R$$ of rank $$n$$

$$det[\lambda I - R] = (-1)^n (r_1 \cdot r_2 \cdots r_n)\prod_{k=1}^n (1-\frac{\lambda}{r_k})$$

$$= (-1)^n e_n(r_1,..,r_n) [ 1 - e_1(1/r_1,..,1/r_n)\lambda + e_2(1/r_1,..) \lambda^2 - ... + e_n(1/r_1,..,1/r_n) \lambda^n]$$

$$= (-1)^n e_n(r_1,..,r_n) E(-\lambda)$$

$$= (-1)^n e_n(r_1,..,r_n) \exp[\ln[\prod_{k=1}^n (1-\frac{\lambda}{r_k})]]$$

$$= (-1)^n e_n(r_1,..,r_n) \exp[-\sum_{j \geq 1}(\frac{1}{r_1^j}+\frac{1}{r_2^j}+ ... +\frac{1}{r_k^j}) \frac{\lambda^j}{j}],$$

$$= (-1)^n e_n(r_1,..,r_n) \exp[\sum_{j \geq 1} -p_j\frac{\lambda^j}{j}]$$

$$= (-1)^n e_n(r_1,..,r_n) \exp[-\sum_{j \geq 1} trace(R^{-j})\frac{\lambda^j}{j}],$$

then

A) Chern roots $$r_k$$ correspond to the zeros of the characteristic polynomial, the eigenvalues $$r_k$$ of $$R$$;

B) Chern classes $$c_k$$ correspond to the elementary symmetric polynomials $$e_k$$ that are the coefficients of the characteristic polynomials in terms of the reciprocals of the Chern roots;

$$c_k = e_k(1/r_1,..,1/r_n),$$

for example,

$$c_1 = e_1(1/r_1,...,1/r_n) = 1/r_1+1/r_2+ ... + 1/r_n,$$

the trace of $$R^{-1}$$, and

$$c_n = e_n(1/r_1,..,1/r_n) =1 / (r_1 \cdot r_2 \cdot ... r_n),$$

the determinant of $$R^{-1}$$,

with the total Chern class polynomial equal to $$E(-\lambda)$$ and the total Chern class, to $$E(-1)$$;

C) Chern characters $$ch_j$$ corresponds to $$j!$$ times the traces of the powers of $$R^{-j}$$, i.e., the power sum symmetric polynomials $$p_j$$ of the reciprocals of the zeros/eigenvalues of $$R$$; that is

$$j! \cdot ch_j(1/r_1,..,1/r_n)= p_j(1/r_1,..,1/r_n) = \frac{1}{r_1^j}+\frac{1}{r_2^j}+..+\frac{1}{r_n^j}$$

with $$ch_0 =$$ the dimension of the vector space under consideration.

Question: What is a standard reference defining the Chern classes, characters, and roots through which I can check my understanding and, if necessary, correct any errors and use as a reference in notes on the topic?

Using the Newton/Waring/Girard identites, or the cycle index polynomials for the symmetric groups (OEIS A036039), the elementary symmetric polynomials, or Chern classes, can be expressed in terms of the power sum polynomials, or Chern characters. Conversely, using the Faber polynomials (A263916), the power sum polynomials, or Chern characters, can be obtained from the Chern classes, or elementary symmetric polynomials. For example,

$$3!e_3(a_1,..,a_n) = 2p_3(a_1,..,a_n) -3p_2(a_1,..,a_n)p_1(a_1,..,a_n) + p_1^3(a_1,..,a_n)$$

and

$$p_3(a_1,..,a_n)= 3 e_3(a_1,..,a_n) - 3 e_1(a_1,..,a_n)e_2(a_1,..,a_n) + e_1^3(a_1,..,a_n).$$

In response to those close votes, this MO-Q by Joe Silverman and attendant comments illustrate the need to at least an introduction of Faber polynomials into a discussion of these topics to fill in a gap of knowledge even among experts in related fields of study (e.g., number theory and elliptic curves). See also Understanding a quip from Gian-Carlo Rota

Some motivation: Zanelli asserts that the following examples involve topological invariants called Chern characteristic classes and Chern-Simons forms

• Sum of exterior angles of a polygon
• Residue theorem in complex analysis
• Winding number of a map
• Poincaré-Hopff theorem (“one cannot comb a sphere”)
• Atiyah-Singer index theorem
• Witten index
• Dirac’s monopole quantization
• Aharonov-Bohm effect
• Gauss’ law
• Bohr-Sommerfeld quantization
• Soliton/Instanton topologically conserved charges

Update 3/30/2021:

I don't have copies of the books mentioned by Debray. However, Tu, in Appendix B. Invariant Polynomials of "Differential Geometry: Connections, Curvature, and Characteristic Classes," denoting by $$X$$ a square matrix, states, "This appendix contains results on invariant polynomials needed in the sections on characteristic classes. We discuss first the distinction between polynomials and polynomial functions. Then we show that a polynomial identity with integer coefficients that holds over the reals holds over any commutative ring with 1. This is followed by the theorem that over the field of real or complex numbers, the ring of invariant polynomials is generated by the coefficients of the characteristic polynomial of -X. Finally, we prove Newton’s identity relating the elementary symmetric polynomials to the power sums. As a corollary, the ring of invariant polynomials over $$R$$ or $$C$$ can also be generated by the trace polynomials." This presentation of the universality under any commutative ring (with identity) is very appealing to me along with the underlying combinatorics of the yoga of symmetric functions. (The phrase "Chern root" is not used in this textbook. I believe "characteristic polynomial" was coined by Weyl.)

• Kobayashi Nomizu, Foundations of Differential geometry, volume 2 Chapter 11 discuss Cheen character Nov 7, 2019 at 3:12
• @PraphullaKoushik, I'll have to steal into a library for that one.. Nov 7, 2019 at 4:21
• Chern roots, classes, and characters are discussed on p. 86 of "Les Houches Lectures on constructing string vacua" by Frederik Denef arxiv.org/abs/0803.1194 Nov 7, 2019 at 4:35
• Or you can ask some one to send scanned copy of that chapter... Can you share from where did you get information about what you have written in last paragraph after the question.. i know some characteristic classes but never saw anything like that before.. Nov 7, 2019 at 4:39
• @PraphullaKoushik, click on Chern classes. It will take you to Wikipedia. Scan down to Constructions and look at the section Via Chern-Weil Theory. The partition polynomials containing the curvature form traces are the cycle index polynomials of OEIS A036039 (see the Lang link for a list of the first ten partition polynomisls). The first three Faber polynomials are often listed to transform from the characters to the classes, but few authors seem aware of the literature on these, or apparently even their name. Relations to the Pontryagin class are given in oeis.org/A231846. Nov 7, 2019 at 4:58

It sounds like, in addition to the references, it would be helpful to disentangle the definitions of Chern roots, Chern classes, and Chern characters. Different mathematicians will have different perspectives; this is mine.

The first thing one defines are Chern classes. Given a complex vector bundle $$E\to X$$, its $$k$$th Chern class is a cohomology class $$c_k(E)\in H^{2k}(X;\mathbb Z)$$. These classes satisfy several nice properties, including:

1. If $$f\colon Y\to X$$ is a map, $$c_k(f^*E) = f^*c_k(E)$$.
2. The total Chern class $$c(E) := c_0(E) + c_1(E) + \dots$$ is multiplicative under direct sum: $$c(E\oplus F) = c(E)c(F)$$.
3. $$c_0(E) = 1$$, and $$c_k(E) = 0$$ if $$k > \mathrm{rank}(E)$$.

There are several different constructions, but you can think of Chern classes as measuring the extent to which $$E$$ is nontrivial, or measuring the curvature of a connection for $$E$$.

A theorem called the splitting principle simplifies some calculations. It tells us that for any complex vector bundle $$E\to X$$, there is a space $$F(E)$$ and a map $$f\colon F(E)\to X$$ such that

1. $$f^*\colon H^*(X; \mathbb Z)\to H^*(F(E); \mathbb Z)$$ is injective, and
2. $$f^*E$$ is a direct sum of line bundles $$L_1,\dotsc,L_r$$.

In particular, $$f^*c(E) = \prod_{i=1}^r c(L_i) = \prod_{i=1}^r (1 + c_1(L_i)).$$ The Chern roots of $$E$$ are $$r_i := c_1(L_i)$$. One reason to care about them is that, since no information was lost upon pulling back to $$F(E)$$, one can prove theorems about Chern classes of $$E$$ by pulling back to $$F(E)$$ and computing with the Chern roots, which are simpler to manipulate. The sum formula above implies the Chern classes are symmetric functions in the Chern roots.

There are many different perspectives on the Chern character; I'll tell you one that I like. The total Chern class behaves nicely under direct sums, but poorly under tensor products. The (total) Chern character $$\mathit{ch}(E)$$ is a characteristic class built out of Chern classes which behaves nicely under direct sums and tensor products, in that $$\mathit{ch}(E\oplus F) = \mathit{ch}(E) + \mathit{ch}(F)$$ and $$\mathit{ch}(E\otimes F) = \mathit{ch}(E)\otimes\mathit{ch}(F)$$.

The standard reference for Chern classes and Chern roots in differential topology (as opposed to algebraic geometry) is either Bott-Tu, Differential forms in algebraic topology, part 4, or Milnor-Stasheff, Characteristic classes. However, I don't think either discusses the Chern character, and I'm not sure what the default reference is for it.

• I'm winging it with "Characteristic classes and K-theory" by Oscar Randal-Williams, Ch. 4, where Chern classes and characters are discussed (dpmms.cam.ac.uk/~or257/teaching/notes/Kthy.pdf) and their connections with the yoga of symmetric functions as well as the axiomatics you present. Nov 7, 2019 at 4:03
• And "CHARACTERISTIC CLASSES" by Andrew Ranicki www2.bc.edu/patrick-orson/indextheory/charclass.pdf. Would you say their perspectives are consistent with yours? Nov 7, 2019 at 4:14
• Yes, to the best of my understanding they're all compatible. Nov 7, 2019 at 4:26
• To quote Wikipedia under the Splitting Principle: Under the splitting principle, characteristic classes for complex vector bundles correspond to symmetric polynomials in the first Chern classes of complex line bundles; these are the Chern classes.// Though the entry is not explicit about which sym polys. See also the quotes in my other comments. Nov 8, 2019 at 14:21
• Just acquired Tu's book. See my update to the question. Mar 31, 2021 at 5:00

Hirzebruch, Friedrich(D-MPI) Topological methods in algebraic geometry. Translated from the German and Appendix One by R. L. E. Schwarzenberger. With a preface to the third English edition by the author and Schwarzenberger. Appendix Two by A. Borel. Reprint of the 1978 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. xii+234 pp. ISBN: 3-540-58663-6

• Thank you. Sections 1.4 and 1.5 of Hirzebruch et al.'s "Manifolds and Modular Forms" (circa 1992) present a discussion of Chern classes in terms of the elementary symmetric polynomials but mentions the Chern character rather than characters. He introduces the Newton identities, including examples of the Faber polynomials, but fails to mention the FP's by name. Of course, the FPs are implicitly defined. Are the remarks in the Wiki link in the Construction section consistent with those of Borel In your ref? Nov 7, 2019 at 9:35
• I am not sure how to compare the remarks in the wiki link with Borel's discussion of the spectral sequence for Dolbeault cohomology of a holomorphic vector bundle on the base of a holomorphic fiber bundle. The idea that Chern classes force sections to vanish on nontrivial cycles works a little differently in the algebraic category, where you can force vanishing or force poles to arise on a nontrivial cycle. So you get more for your money in the algebraic category. Nov 8, 2019 at 9:28